For a hyperbolic $3$-manifold $M$ with a torus boundary component,
all but finitely many Dehn fillings yield hyperbolic $3$-manifolds.
In this paper, we will focus on the situation where
$M$ has two exceptional Dehn fillings: an annular filling and a toroidal filling.
For such a situation, Gordon gave an upper bound of $5$ for the distance between such slopes.
Furthermore, the distance $4$ is realized only by two specific manifolds, and $5$
is realized by a single manifold.
These manifolds all have a union of two tori as their boundaries.
Also, there is a manifold with three tori as its boundary which realizes the distance $3$.
We show that if the distance is $3$ then the boundary of the manifold consists of at most three tori.